A147626 Octo-factorial numbers (5).

1, 6, 84, 1848, 55440, 2106720, 96909120, 5233092480, 324451733760, 22711621363200, 1771506466329600, 152349556104345600, 14320858273808486400, 1460727543928465612800, 160680029832131217408000, 18960243520191483654144000, 2388990683544126940422144000

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..325

Crossrefs

Cf. A034908, A045755, A048994, A049210, A051189, A052714, A132393, A147625.

Programs

Magma

[n le 1 select 1 else (8*n-10)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

Mathematica

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,8}];lst
Table[8^(n-1)*Pochhammer[3/4, n-1], {n,40}] (* G. C. Greubel, Oct 21 2022 *)

SageMath

[8^(n-1)*rising_factorial(3/4, n-1) for n in range(1,40)] # G. C. Greubel, Oct 21 2022

Formula

a(n+1) = Sum_{k=0..n} A132393(n,k)*6^k*8^(n-k). - Philippe Deléham, Nov 09 2008

a(n) = (-2)^n*Sum_{k=0..n} 4^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012

G.f.: 2*x/G(0), where G(k) = 1 + 1/(1 - 2*x*(8*k+6)/(2*x*(8*k+6) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

From G. C. Greubel, Oct 21 2022: (Start)

a(n) = 8^n * Pochhammer(n, 3/4) = -2^(3*n-1) * Pochhammer(n, -1/4).

a(n) = (8*n - 10)*a(n-1). (End)

Sum_{n>=1} 1/a(n) = 1 + (e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). - Amiram Eldar, Dec 20 2022